{
  "id": "2007.11254",
  "version": "v1",
  "published": "2020-07-22T08:19:39.000Z",
  "updated": "2020-07-22T08:19:39.000Z",
  "title": "Topological groups with invariant linear spans",
  "authors": [
    "Eva Pernecká",
    "Jan Spěvák"
  ],
  "comment": "5 pages",
  "categories": [
    "math.GN"
  ],
  "abstract": "Given a topological group $G$ that can be embedded as a topological subgroup into some topological vector space (over the field of reals) we say that $G$ has invariant linear span if all linear spans of $G$ under arbitrary embeddings into topological vector spaces are isomorphic as topological vector spaces. For an arbitrary set $A$ let $\\mathbb{Z}^{(A)}$ be the direct sum of $|A|$-many copies of the discrete group of integers endowed with the Tychonoff product topology. We show that the topological group $\\mathbb{Z}^{(A)}$ has invariant linear span. This answers a question of D. Dikranjan et al. in positive. We prove that given a non-discrete sequential space $X$, the free abelian topological group $A(X)$ over $X$ is an example of a topological group that embeds into a topological vector space but does not have invariant linear span.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-22T08:19:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46A99",
      "22A99"
    ],
    "keywords": [
      "invariant linear span",
      "topological vector space",
      "tychonoff product topology",
      "non-discrete sequential space",
      "free abelian topological group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}